We use generic bricks in the study of arbitrary biserial algebras. For a biserial algebra \(\Lambda \) of rank n over an algebraically closed field k, we show that \(\Lambda \) is brick-infinite if and only if it admits a generic brick, which we further prove to be the case if and only if there exists an infinite family of bricks of length d, for some \(2\le d\le 2n\) . Consequently, we obtain an algebro-geometric realization of \(\tau \) -tilting finiteness of biserial algebras: \(\Lambda \) is \(\tau \) -tilting finite if and only if, for each dimension vector \(\underline{d}\) , there are only finitely many orbits of bricks in the representation variety \(\mathop {\textrm{mod}}\limits (\Lambda , \underline{d})\) . Our results rely on our full classification of minimal brick-infinite biserial algebras in terms of quivers and relations, seen as the modern analogue of the classification of minimal representation-infinite (special) biserial algebras, given by Ringel. We show that if \(\Lambda \) is a minimal brick-infinite biserial algebra, then \(\Lambda \) is gentle and admits exactly one generic brick. In this case, we describe the spectrum of \(\Lambda \) and prove that it is similar to that of a tame hereditary algebra. In other words, \(\mathop {\textrm{Brick}}\limits (\Lambda )\) is the disjoint union of a unique generic brick with a countable infinite set of bricks of finite lengths, and a family of bricks of length d parameterized by k. Our work strengthens and generalizes some earlier results on \(\tau \) -tilting finiteness of gentle algebras and special biserial algebras, respectively treated by Plamondon and Schroll-Treffinger-Valdivieso.
